On Nonnegative Signed Domination in Graphs and its Algorithmic Complexity

Abstract

<span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;"></span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">Let <em style="mso-bidi-font-style: normal;">G</em></span><em style="mso-bidi-font-style: normal;"><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US"> </span></em><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">=</span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US"> </span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">(<em style="mso-bidi-font-style: normal;">V</em>,</span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US"> </span><em style="mso-bidi-font-style: normal;"><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">E</span></em><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">) be a simple graph with vertex set <em style="mso-bidi-font-style: normal;">V</em> and edge set <em style="mso-bidi-font-style: normal;">E</em>. </span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">A function <em style="mso-bidi-font-style: normal;">f</em></span><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US"> from </span><em style="mso-bidi-font-style: normal;"><span style="color: black; font-family: "Times New Roman"; font-size: 10pt; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;" lang="EN-US">V</span></em><span style="color: black; font-fam...

Lower bounds on nonnegative signed domination parameters in graphs

“…Theorem D. [8] Let K n be a complete graph. Then γ N N s (K n ) = 0 when n is even and γ N N s (K n ) = 1 when n is odd.…”

“…The concept of the signed k-subdomination number was introduced and studied by Cockayne and Mynhardt [3]. A nonnegative signed dominating function (NNSDF) of G defined in [8] as a function f :…”

Remarks on the Complexity of Non-negative Signed Domination

“…In 2013, Huang, Li, and Feng [15] introduced a variation of signed domination called non-negative signed domination. They gave formulas to compute the nonnegative signed domination numbers for paths, stars, wheels, spiders, complete graphs, and complete equally bipartite graphs, and presented some lower bounds for non-negative signed domination number in terms of the order and the maximum and minimum degrees of a graph.…”

“…Proof: Note that the non-negative signed domination problem is NP-complete for bipartite planar graphs and doubly chordal graphs. By using the arguments similar to those for proving the NP-completeness of the zero nonnegative signed domination problem on bipartite graphs and chordal graphs in [15], we can prove the NPcompleteness of the zero non-negative signed domination problem on bipartite planar graphs (respectively, doubly chordal graphs) by a polynomial-time reduction from the non-negative signed domination problem on bipartite planar graphs (respectively, doubly chordal graphs).…”

On the nonnegative signed domination numbers in graphs

“…The nonnegative signed domination number was introduced by Huang et al [3]. In their paper, they determined the exact values of this parameter for some classes of graphs.…”